Uncertainty quantification (UQ) is vital for trustworthy deep learning, yet existing methods are either computationally intensive, such as Bayesian or ensemble methods, or provide only partial, task-specific estimates, such as single-forward-pass techniques. In this paper, we propose a post-hoc single-forward-pass framework that jointly captures aleatoric and epistemic uncertainty without modifying or retraining pretrained models. Our method applies \emph{Split-Point Analysis} (SPA) to decompose predictive residuals into upper and lower subsets, computing \emph{Mean Absolute Residuals} (MARs) on each side. We prove that, under ideal conditions, the total MAR equals the harmonic mean of subset MARs; deviations define a novel \emph{Self-consistency Discrepancy Score} (SDS) for fine-grained epistemic estimation across regression and classification. For regression, side-specific quantile regression yields prediction intervals with improved empirical coverage, which are further calibrated via SDS. For classification, when calibration data are available, we apply SPA-based calibration identities to adjust the softmax outputs and then compute predictive entropy on these calibrated probabilities. Extensive experiments on diverse regression and classification benchmarks demonstrate that our framework matches or exceeds several state-of-the-art UQ methods while incurring minimal overhead. Our source code is available at https://github.com/zzz0527/SPC-UQ.

Wearable photoplethysmography (PPG) is embedded in billions of devices, yet its optical waveform is easily corrupted by motion, perfusion loss, and ambient light, jeopardizing downstream cardiometric analytics. Existing signal-quality assessment (SQA) methods rely either on brittle heuristics or on data-hungry supervised models. We introduce the first fully unsupervised SQA pipeline for wrist PPG. Stage 1 trains a contrastive 1-D ResNet-18 on 276 h of raw, unlabeled data from heterogeneous sources (varying in device and sampling frequency), yielding optical-emitter- and motion-invariant embeddings (i.e., the learned representation is stable across differences in LED wavelength, drive intensity, and device optics, as well as wrist motion). Stage 2 converts each 512-D encoder embedding into a 4-D topological signature via persistent homology (PH) and clusters these signatures with HDBSCAN. To produce a binary signal-quality index (SQI), the acceptable PPG signals are represented by the densest cluster while the remaining clusters are assumed to mainly contain poor-quality PPG signals. Without re-tuning, the SQI attains Silhouette, Davies-Bouldin, and Calinski-Harabasz scores of 0.72, 0.34, and 6173, respectively, on a stratified sample of 10,000 windows. In this study, we propose a hybrid self-supervised-learning--topological-data-analysis (SSL--TDA) framework that offers a drop-in, scalable, cross-device quality gate for PPG signals.

TabPFN v2 achieves better results than tree-based models on several tabular benchmarks, which is notable since tree-based models are usually the strongest choice for tabular data. However, it cannot handle more than 10K context tokens because transformers have quadratic computation and memory costs. Unlike existing approaches that rely on context compression, such as selecting representative samples via K-nearest neighbors (KNN), we introduce a tiled-block strategy to compute attention within the TabPFN framework. This design is compatible with standard GPU setups and, to the best of our knowledge, is the first to enable TabPFN to process long contexts without any pre-processing. We demonstrate the effectiveness of our approach on the standard TabArena benchmark, with code available at chunk tabpfn.

Discrete Bayesian networks (DBNs) provide a broadly useful framework for modeling dependence structures in multivariate categorical data. There is a vast literature on methods for inferring conditional probabilities and graphical structure in DBNs, but data sparsity and parametric assumptions are major practical issues. In this article, we detail a comprehensive Bayesian framework for learning DBNs. First, we propose a hierarchical prior for the conditional probabilities that enables complicated interactions between parent variables and stability in sparse regimes. We give a novel Markov chain Monte Carlo (MCMC) algorithm utilizing parallel Langevin proposals to generate exact posterior samples, avoiding the pitfalls of variational approximations. Moreover, we verify that the full conditional distribution of the concentration parameters is log-concave under mild conditions, facilitating efficient sampling. We then propose two methods for learning network structures, including parent sets, Markov blankets, and DAGs, from categorical data. The first cycles through individual edges each MCMC iteration, whereas the second updates the entire structure as a single step. We evaluate the accuracy, power, and MCMC performance of our methods on several simulation studies. Finally, we apply our methodology to uncover prognostic network structure from primary breast cancer samples.

Scientific machine learning increasingly uses spectral methods to understand physical systems. Current spectral learning approaches provide only point estimates without uncertainty quantification, limiting their use in safety-critical applications where prediction confidence is essential. Parametric matrix models have emerged as powerful tools for scientific machine learning, achieving exceptional performance by learning governing equations. However, their deterministic nature limits deployment in uncertainty quantification applications. We introduce Bayesian parametric matrix models (B-PMMs), a principled framework that extends PMMs to provide uncertainty estimates while preserving their spectral structure and computational efficiency. B-PMM addresses the fundamental challenge of quantifying uncertainty in matrix eigenvalue problems where standard Bayesian methods fail due to the geometric constraints of spectral decomposition. The theoretical contributions include: (i) adaptive spectral decomposition with regularized matrix perturbation bounds that characterize eigenvalue uncertainty propagation, (ii) structured variational inference algorithms using manifold-aware matrix-variate Gaussian posteriors that respect Hermitian constraints, and (iii) finite-sample calibration guarantees with explicit dependence on spectral gaps and problem conditioning. Experimental validation across matrix dimensions from 5x5 to 500x500 with perfect convergence rates demonstrates that B-PMMs achieve exceptional uncertainty calibration (ECE < 0.05) while maintaining favorable scaling. The framework exhibits graceful degradation under spectral ill-conditioning and provides reliable uncertainty estimates even in near-degenerate regimes. The proposed framework supports robust spectral learning in uncertainty-critical domains and lays the groundwork for broader Bayesian spectral machine learning.

We prove ratio-consistency of the jackknife variance estimator, and certain variants, for a broad class of generalized U-statistics whose variance is asymptotically dominated by their Hájek projection, with the classical fixed-order case recovered as a special instance. This Hájek projection dominance condition unifies and generalizes several criteria in the existing literature, placing the simple nonparametric jackknife on the same footing as the infinitesimal jackknife in the generalized setting. As an illustration, we apply our result to the two-scale distributional nearest-neighbor regression estimator, obtaining consistent variance estimates under substantially weaker conditions than previously required.

The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization and instead follows the formal Riemannian geometry of Wasserstein space. In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements, with barycentric projections of optimal transport plans used in place of Monge maps when the latter do not exist. This yields a geometry-aware particle-flow update that preserves sharp features of the Wasserstein barycenter while remaining computationally tractable. We establish theoretical guarantees, including consistency of barycentric projections, monotone descent and convergence to stationary points, stability with respect to perturbations of the inputs, and resolution consistency as the number of atoms increases. Empirical studies on averaging probability distributions, Bayesian posterior aggregation, image prototypes and classification, and large-scale clustering demonstrate accuracy and scalability of the proposed particle-flow approach, positioning it as a principled alternative to both linear programming and regularized solvers.

Many human tasks benefit from prior experience when that experience is related to the task at hand. This phenomenon, whereby knowledge from previous tasks is transferred to new ones, has motivated the machine learning technique known as transfer learning. From a statistical perspective, consider the problem of analyzing a regression relationship when the available data are limited. Transfer learning (Torrey and Shavlik (2010)), one of the most widely used techniques in machine learning, can provide a solution. In this framework, one typically leverages related estimates obtained from large but non-identically distributed auxiliary samples, and then refines these estimates to obtain improved estimators from the smaller target sample. Transfer learning has been shown to be effective in a wide range of real-world applications, including computer vision (Kolesnikov et al. (2020); Bu et al. (2021)), natural language processing (Lee et al. (2020); Yuan et al. (2020)), and bioinformatics (Vorontsov et al. (2024); Gao and Cui (2020)), among others. Recently, the theoretical properties of transfer-learned estimators have been extensively investigated across a range of statistical problems.

Estimating density ratios is a fundamental problem in machine learning, but existing methods often trade off accuracy for efficiency. We propose \textit{Interval-annealed Secant Alignment Density Ratio Estimation (ISA-DRE)}, a framework that enables accurate, any-step estimation without numerical integration. Instead of modeling infinitesimal tangents as in prior methods, ISA-DRE learns a global secant function, defined as the expectation of all tangents over an interval, with provably lower variance, making it more suitable for neural approximation. This is made possible by the \emph{Secant Alignment Identity}, a self-consistency condition that formally connects the secant with its underlying tangent representations. To mitigate instability during early training, we introduce \emph{Contraction Interval Annealing}, a curriculum strategy that gradually expands the alignment interval during training. This process induces a contraction mapping, which improves convergence and training stability. Empirically, ISA-DRE achieves competitive accuracy with significantly fewer function evaluations compared to prior methods, resulting in much faster inference and making it well suited for real-time and interactive applications.

Ensuring consistent quality in vacuum thermoforming presents challenges due to variations in material properties and tooling configurations. This research introduces a vision-based quality control system to predict and optimise process parameters, thereby enhancing part quality with minimal data requirements. A comprehensive dataset was developed using visual data from vacuum-formed samples subjected to various process parameters, supplemented by image augmentation techniques to improve model training. A k-Nearest Neighbour algorithm was subsequently employed to identify adjustments needed in process parameters by mapping low-quality parts to their high-quality counterparts. The model exhibited strong performance in adjusting heating power, heating time, and vacuum time to reduce defects and improve production efficiency.